Can an algorithm be faster than O(1)?Your analog computation is still $O(1)$ due to relativistic considerations. From a physical perspective, since the observable universe has finite usable energy, it will always take time bounded away from zero to output the result of a non-null algorithm.

Should I write out stuff?@Alex J: I interpret the quote as follows: For each theorem that Bourgain proved, he did not need to do a computation to convince himself that it was true. Often, there are structural methods outside computation that you can use to make guesses.